Chapter 2 – Part 2

Chapter 2 – Part 2 Liner System Review, DFT & FFT Updated:2/23/15

Outline •  Review of linear systems •  Sampling theorem •  Fast Fourier Transform

Linear Time Invariant System (LTIS) - 1 L is a Linear Operation

Example: y(t) = t – 3 Is a linear time invariant system

Linear Time Invariant System (LTIS) - 2

h(t)

δ(t)

δ(t-7t) Δt

tà n=1 2 3 4 5 6 7 8 .... N

Linear Time Invariant System (LTIS) - 3

This is called the convolution integral!

Linear Time Invariant System (LTIS) - 4

Example: Linear Time Invariant System (LTIS) - 5

power transfer function (or power gain) of the system

Example: RC Low-Pass Filter Characterization

See Fourier Pair Table (Exponential one-sided)

10log(|H(f)|^2)=0dBßà 1.0

10log(|H(f)|^2)=10log (0.5)=-3dBßà 0.5

When f=foà G(fo)=0.5à-3dB attenuation

Distortionless Transmission -1 •  An LTI system is termed distortionless if it introduces the same attenuation to all spectral components and offers linear phase response over the frequency band of interest:

Ho is the gain (or attenuation!) If Ho is unity then there is no lossà Lossless system We refer to to as the Td or time delay

Distortionless Transmission -2

Note that the phase response is a linear function of frequency in LTI! Group delay: refers to time delay that difference spectral components experience!

Distortionless Transmission -3 •  The phase delay of an LTI system is defined as

•  For a LTI system

(from before)

Is the Output of an RC Filter Distortionless? Remember, for RC filter:

-

Introducing both amplitude and phase distortion! …see next

Is the Output of an RC Filter Distortionless? Amplitude distortion if the amplitude response is not flat

Range of frequencies ( 2B •  Δf is frequency resolution = 1/T •  f represents the frequency points = n/T ; n = [0,1,2, N-1]

Sampled Windowed waveform and its magnitude spectrum – fs=1/dt

Periodic Sampled Windowed waveform and its magnitude spectrum – fs=1/dt=N/T & dt=T/N (or Period T = N.dt) & fo=1/To

X(n) is the DFT

Using FFT to find the DFT - MATLAB Example M = 7; N = 2^M; % Using zero padding n = 0:1:N-1; T = 10; % period dt = T/N; t = n*dt;

% sampling period % simulation time

Tend = 1

T=10

Zoomed to f = [0, 4]

% Creating time waveform % w=Your waveform! % Calculating FFT W = dt*fft(w); f = n/T; plot(t,w); plot(f,abs(W); plot(f,180/pi*angle(W));

Pos. Freq.

Neg. Freq.

Using DFT to Compute the Fourier Series w

w

Example (MATLAB Implementation)

We use the DFT (FFT) to approximate the spectrum continuous W(f) & evaluate the complex Fourier series coefficients cn

Example (MATLAB Implementation) fo=10

Magnitude Spectrum: |Cn|

10Hz

70Deg. @ 10Hz

Note f=n*fo=10

References •  Leon W. Couch II, Digital and Analog Communication Systems, 8th edition, Pearson / Prentice, Chapter 1 •  Electronic Communications System: Fundamentals Through Advanced, Fifth Edition by Wayne Tomasi – Chapter 2 (https://www.goodreads.com/book/show/209442.Electronic_Communications_System)